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State if the following is true or false.

In an isosceles ΔABC with AC=BC, the base AB is produced both ways to P and Q equally such that $AP × BQ = AC 2$ then $: Δ ACP ~ Δ BQC$

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True

False

answer is A.

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Detailed Solution

Given figure is,
Question Image

We know,
SAS similarity states that if two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar.
It is given that

Δ A B C

is an isosceles triangle and the sides AC=BC.
As the sides AC and BC are equal, the corresponding angles of the sides are also equal.

∠ C A B = ∠ C B A ⇒ 180 ° − ∠ C A B = 180 ° − ∠ C B A ⇒ ∠ C A P = ∠ C B Q

It is given that AC=BC.
Then, substitute BC for AC in the equation,

\frac{BQ}{BC} = \frac{AC}{AP}

\Rightarrow

A P A C = B C B Q

…………(1)
Also,
∠CAB=∠CBA

\Rightarrow 180 ° - ∠ CAB = 180 ° - ∠ CBA

\Rightarrow ∠ CAP = ∠ CBQ

……….(2)
From (1) and (2),

Δ A C P

and

Δ B Q C

are similar by SAS similarity.
Therefore, the statement is true.
Option 1 is correct.

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